Multi-Unit Demand Auctions with Synergies:
Behavior in Sealed-Bid versus Ascending-Bid Uniform-Price Auctions*
John H. Kagel and Dan Levin
Department of Economics
The Ohio State University
Revised July 12, 2004
* This research was supported by a grant from the National Science Foundation. This research
has benefitted from helpful discussions with Larry Ausubel, Peter Cramton, Elena Katok, John
Ledyard, Howard Marvel, Jim Peck, Dave Porter and participants at the ESA conference and the
Ohio State microeconomic’s seminar. We have received excellent research assistance from HuiJang Chang, Guillaume Frechette, Jun Han, Scott Kinross, and Carol Kraker Stockman and
editorial assistance from Jo Ducey and Mark Owens.
Corresponding author: John H. Kagel, Department of Economics, 410 Arps Hall, 1945 North
High Street, Columbus, OH 43210-1172, USA. [email protected]. 614-292-4812 (phone). 614292-4192 (Fax).
Abstract
We construct a model of bidding with synergies and solve it for both open outcry and sealed-bid
uniform price auctions. The model is simple enough to allow for direct interpretations of the
experimental data, while still maintaining the essential behavioral forces involved in auctions
with synergies: (1) A demand reduction force resulting from the monopsony power that bidders
with multiple-unit demands have when synergies are relatively inconsequential and (2) Bidding
above standalone values in order to capture significant complementarities between units. The
latter creates a potentially important behavioral force - the “exposure problem” - as bidders may
win only parts of a package and earn negative profits. Bidding outcomes are closer to
equilibrium in clock compared to sealed-bid auctions. However, there are substantial and
systematic deviations from equilibrium, with patterns of out-of-equilibrium play differing
systematically between the two auction formats. These patterns of out-of-equilibrium play are
analyzed, along with their effects on revenue and efficiency.
JEL Classification: D44, C92, D80.
Keywords: auction, synergies, exposure problem, experiment.
The FCC spectrum auctions have reinvigorated theoretical and empirical research on
auctions in efforts to better understand the effects of different auction institutions when
individual bidders demand multiple units of a given commodity. One line of research has
focused on the performance of auctions with uniform-price rules, where all winning bids pay the
same highest rejected bid.1 It is well known by now that in such auctions, when valuations are
non-increasing, bidders have an incentive to reduce demand on some of their units in order to
exploit the monopsony power they have when demanding multiple units. This strategy may
result in winning fewer units, but when it does, it also reduces the price on units earned. (See, for
example, Ausubel and Cramton, 1996 and Englebrecht-Wiggans and Kahn, 1998.) Demand
reduction reduces economic efficiency and revenue relative to a full demand revelation.
Experimental, and quasi-experimental research confirms that the demand reduction incentives
are reasonably transparent and practiced even by relatively naive bidders (Kagel and Levin,
2001; List and Lucking-Reily, 2000). Further, experiments comparing sealed-bid auctions with
ascending-price clock auctions reveal that although both auctions have the same normal form
game representation, bidding is significantly closer to equilibrium in the clock auction,
suggesting there are behavioral elements not fully captured in the theory (Kagel and Levin,
2001).
Uniform-price auctions which involve synergies, or complementarities, provide
additional incentives and generate radically different bidding strategies than the same auctions
without synergies. Synergies create an opposite incentive to the demand reduction force:
aggressive bidding in order to acquire desired packages with their super additive value. Further,
in sealed-bid uniform-price auctions, that do not permit package bids, the existence of synergies
commonly dictates submitting bids above the standalone values for individual units in order to
increase the probability of winning a package with its super additive value. However, this
3
strategy is risky since if a bidder fails to acquire the whole package and wins only parts instead,
she is likely to earn negative profits. Thus, in addition to the competing equilibrium incentives,
an important “behavioral” force may affect bidding as well: Depending on the size of the
potential loss, and risk preferences, bidders may refrain from such aggressive bidding in order to
avoid exposure to such losses, despite the benefits of doing so (Bykowsky et al., 1995; Ausubel
et al., 1997; Rothkopf et al., 1998). This avoidance has been referred to as the “exposure
problem,” a serious concern in some quarters at least, in designing auctions in the absence of that
package bids.2
The present paper reports the results of an experiment in a highly simplified auction
environment designed to maintain the essential richness of the economic and behavioral forces
present in multi-unit demand auctions with synergies. We first construct a tractable model of
bidding with synergies and solve it for both open outcry and sealed-bid uniform price auctions
(highest losing bid determines the price paid). The model is different from any existing model of
auctions with synergies (being closest in structure to the model developed by Krishna and
Rosenthal, 1996; see below): It is simple enough to allow for direct interpretations of the
experimental data, while still maintaining the essential behavioral forces involved in auctions
with synergies. We compare outcomes of sealed-bid and ascending-bid (English-clock) uniformprice auctions. Under our design, the net effect of the demand reduction force and the synergy
force is that in equilibrium: (1) at lower valuations, the demand reduction force dominates so that
bidders shave their bids on marginal units, (2) at the highest valuations the synergy force
dominates so that bidders “go for it,” bidding high enough to insure winning the items, and (3) at
middle valuations the two forces are at peak tension and are counterbalancing each other, with
bidding above value (but short of “going for it”) in the sealed-bid auctions and “going for it,”
conditional on rivals’ observed dropout prices, in the clock auctions. The exposure problem
4
works against the synergy effect, being most prominent at middle valuations when bidding above
standalone values but short of going for it, and when valuations are such that going for it does
not insure earning a positive profit.
Under our experimental design a human subject demanding two units of a commodity
competes against different numbers of rivals demanding a single unit of the commodity in a
uniform-price auction. Single-unit buyers have a dominant strategy, bidding their value, and are
played by computers.3 The standalone values for both items are the same for the human bidder,
vh, but earning both units generates three times the standalone value (3vh). With independent
private values drawn from a uniform distribution and with supply of two units, the equilibrium
predictions for the “large” bidder correspond to the three regions characterized above. Thus, the
experimental design is simple enough to yield equilibrium predictions while still maintaining the
tension between the demand reduction and synergy forces. The design also abstracts away from
the strategic uncertainties inherent in interactions between human bidders (e.g., problems of
learning best responses given rivals’ out-of-equilibrium bids). Finally, the experimental design
allows us to employ a limited number of values for the human bidders without distorting the
equilibrium predictions. We exploit this by limiting the number of standalone values in each
experimental session to three, with a number of replications at each value, thereby providing
bidders with more systematic, and easier to process, feedback in this relatively complicated
bidding environment. The standalone values employed span the strategy space and induce
maximum differences in strategic behavior between the sealed-bid and clock auctions, while
providing a number of replications at each value against which to evaluate behavior.
We do not expect bidders to be able to calculate and respond precisely to the fine cut-off
points associated with such a complex auction environment. Thus, we focus on the following
questions: Are bidders sensitive to the tradeoffs inherent in uniform-price auctions with
5
synergies? Do they behave differently in cases where the demand reduction force dominates
compared to cases where the synergy bonus is strong enough to dominate? What role, if any,
does the exposure problem play in bidding? What are the nature of deviations from optimal
bidding strategies, and are there systematic differences in the patterns of deviations between the
two types of auctions studied? Are there public policy implications resulting from any systematic
deviations from optimal bidding?
There has been some experimental work on multi-unit demand auctions with synergies.
The work falls roughly into two major categories: First, “test bed” experiments designed to
explore the effects of different auction rules for public policy purposes; in particular, to provide
data and insights into urgent problems arising in the design of the FCC spectrum auctions, with an
emphasis on the efficiency of alternative auction mechanisms (see, for example, Ledyard et al.,
1997 and Plott, 1997). In most cases these experiments explore situations for which theory has
little to say, with a strong focus on environments where package bidding might play a role in
achieving efficient resource allocations (Ledyard et al., 1997). Our experimental environment
represents a tremendous simplification relative to these experiments as we reduce the complexity
of the game for multi-unit demand bidders to a well-defined decision problem. This clearly
represents “backtracking” relative to the effort in these experiments to capture the complexity of
the demand structure underlying the FCC auctions. However, our research strategy has several
key advantages resulting from this “backtracking”: (1) We are able to solve for the optimal
bidding strategy which provides us with a clean theoretical benchmark against which to evaluate
behavior, while still preserving many of the essential economic tradeoffs that multi-unit demand
bidders face in more complicated settings with synergies, and (2) We have a large number of
observations against which to evaluate behavior. As a result of these advantages we are able to
examine in detail, against a clean theoretical benchmark, many of the important behavioral issues
6
involved in auctions with synergies. In contrast, in the test bed experiments, as Ledyard et al.
note, “No careful theoretical analysis or experimental design was followed, nor could one be,
given the urgency of the situation.” (Ledyard et al., 1997, p. 641) Unfortunately, the large
differences in underlying economic structure between our experiment and these test bed
experiments makes direct comparisons of results problematic.
The second major category of experimental work on auctions with synergies are
experiments designed to explore the feasibility of Vickrey auctions with package bids (Isaac and
James, 2000; Brenner and Morgan, 1997). Although the emphasis in both these experiments is
on the Vickrey auctions, they employ control treatments that are reasonably close in spirit to our
experimental design: Brenner and Morgan employ a simultaneous ascending price auction
designed to mimic FCC procedures. Isaac and James employ a simultaneous sealed bid second
price auction similar to the one described in Krishna and Rosenthal (1996). In both cases the role
of these control treatments is to provide a reference point against which to evaluate the potential
efficiency gains associated with the Vickrey auction. There is virtually no analysis of behavior
within the control treatments against which to compare our results: Brenner and Morgan limit
their analysis to a single paragraph pointing out that they can reject a hypothesis of demand
revelation in the simultaneous ascending price auction in favor of bid shaving. Isaac and James
(p. 52) limit their analysis to an endnote pointing out that they saw cases in which high value
bidders lost the auction because they bid too low on individual items and other cases where they
bid over their valuations. Thus, there is little basis for comparing our experimental results to
these control conditions, and we do not address the issue of package bids or the question of
Vickrey auctions here.
Ausubel et al. (1997) and Morten and Spiller (1996) examine license interdependencies in
some of the early FCC spectrum auctions. The FCC auctions involve heterogenous goods
7
auctioned off simultaneously in a number of separate markets, in contrast to the single market
with homogenous goods and synergies explored here (see below).
The auction model underlying our experiment is similar to one developed in Krishna and
Rosenthal (1996) to explore simultaneous sealed-bid auctions with synergies. In both cases there
is a single bidder demanding two units competing against a number of rivals demanding a single
unit. The primary difference between the two models is that in the Krishna and Rosenthal model
the bidder demanding multiple units competes in two separate second-price auctions against n
single-unit demand bidders in each market.4 In other words, in our model the two goods are
perfect substitutes and sold together in a single uniform-price auction. In Krishna and Rosenthal
the two goods are imperfect substitutes and sold in two separate second-price auctions. As a
result, there is no demand reduction force present in their model as there is in ours. However, in
regions of our experimental design where the synergy force dominates the demand reduction
force, the two models make remarkably similar predictions: Single-unit bidders always bid their
value. When bidding above value, the bidder with increasing returns always bids the same on
both units, with bids increasing in the valuation drawn.5 Once the valuation is high enough, there
is a discontinuous jump in the bid function, so that the bidder with increasing demand “goes for
it.” Further, bids of the multi-unit demand bidder are weakly decreasing with more competition,
as they are in our model. Finally, Krishna and Rosenthal do not extend their analysis to
ascending-bid clock auctions as we do here.
Our experiment yields a number of basic insights: Bidders are always closer to optimal
bidding strategies in a clock compared to sealed-bid auctions. Bidders are responsive to the
underlying economic forces present in the auction even though there is considerable out-ofequilibrium play. Further, out-of-equilibrium play differs substantially and systematically
between the two institutions. In the sealed-bid auctions there is a clear tendency for bidders to
8
overbid at low values and underbid at high values. In contrast, in the clock auctions, absent
secure, positive
expected profits, there is a general reluctance to bid above value when optimality requires it,
consistent with the “exposure problem.” At least part of this differential sensitivity to the
exposure problem in clock compared to sealed-bid auctions results from the obvious ability to
stop the bidding and assure a positive profit in the clock auctions. This is indicative of a clear
presentation format effect, an outcome observed in other auction settings as well (Kagel et al.,
1987; Kagel and Levin, 2001). As a result, the clock auction fails to improve efficiency relative
to the sealed-bid auctions where the theory predicts it should, and the sealed-bid auctions generate
uniformly higher revenue.
The plan of the paper is as follows: Section 1 develops the theoretical predictions for both
ascending-bid and sealed-bid auctions. The experimental design is outlined in Section 2 along
with the theoretical predictions specific to the experimental design. Results of the experiment are
reported in Section 3. We close with a brief summary and discussion of our major results.
1. Theoretical Predictions
We investigate bidding in IPV auctions with (n+1) bidders and m indivisible, identical
objects for sale, where n $ m. Each bidder i (i = 1, ... , n) demands a single unit of the good,
placing a value vi on the good. These bidders are indexed by their values so that v1$v2$, ...,$vn.
Bidder h, the (n+1)th bidder, demands two units of the good, with the value of each unit by itself
equal to vh. However, earning two units generates synergies so that the value of winning both
units = 2vh + "vh. Bidders’ values were independent and identical draws (iid) from a uniform
distribution on the interval [0,V] for all bidders. In what follows we work with m = 2, " =1, as
these are the values employed in the experiment, and analyze behavior within the unit interval (V
= 1).6 For both sealed-bid and clock auctions there are three bidding regions, with distinctly
9
different bidding strategies for the human bidder in each region, which are discussed in detail
below.
A. Sealed-bid auctions: In the sealed-bid auction, each bidder simultaneously submits sealed bids
for each of the units demanded. These are ranked from highest to lowest, with the m highest bids
each winning an item and paying a price equal to the m+1 highest bid.
For bidders i = 1, ... , n, demanding a single unit of the commodity, there is a dominant
strategy (the same as in the familiar single-unit Vickrey auction) to bid their value, vi . Bidder h
demands two units of the commodity and submits two bids, b1 and b2 for unit one and two
respectively. Without loss of generality assume that b1 $ b2. The optimal bidding strategy varies
dramatically with vh, since this directly affects the tradeoffs between incentives promoting
demand reduction and those promoting bidding above vh in order to benefit from the synergy
bonus as a result of winning both units. There are three regions of interest. (See the top panel of
Figure 1 which provides a characterization of the bid patterns over the unit interval for n = 3 and
5, the number of single unit bidders employed in our experimental design. Appendix A provides
the derivation of the equilibrium bidding strategy for the sealed-bid auction.)
[Insert Figure 1 here]
Region 1: For vh < ½, b1 = vh and b2 = 0. (For vh = ½, b1 = b2 = ½. ) The synergy value is not
large enough to overcome the demand reduction incentive. Here, bidders are better off exercising
drastic demand reduction, giving up any chance of winning two units.
Region 2: For ½ < vh < v(n), vh < b1 = b2 < 1, where v(n) is defined as the upper bound of the
interval for region 2, and is a function of the number of single unit bidders h is competing against
(see Appendix A). That is, the optimal bidding strategy calls for submitting two equal bids above
vh, but not high enough to assure winning both units. Both the size of this interval and how much
to bid above vh varies with n, with a wider interval and more aggressive bidding the smaller n is.
10
Region 3: For vh $ v(n), b1 = b2 $ 1. That is, the optimal bidding strategy is to “go for it,” bidding
high enough to insure winning both units as the expected value of winning both units, even at the
maximal bid price of V (=1) yields positive expected profit given the synergy bonus. In contrast,
in region 2, values are not high enough to insure positive expected profit at the maximum bid
price, so that bidders do not go for it in region 2.
Risk aversion has no effect on bids in region 1: b1 = vh is based on a dominance argument,
and risk aversion will lower b2 in region 1, unless b2 = 0 as in our design (see Appendix A). For vh
high enough in region 3, h is assured of earning nonnegative profits even paying the maximum
possible price (p = 1), so that risk preferences play no role here as well. For region 2 and the
remainder of region 3, the requirement that b1 = b2 is satisfied regardless of risk preferences, as it
is based on a dominance argument as well. (b1 = b2 $vh dominates b1* > b2 $vh since the higher b1*
only matters when winning one unit and regretting it as the price must be above vh.) However, the
level of bids in this interval will, in general, be affected by risk aversion, as risk averse bidders
require a risk premium to try for both units unless they are assured of earning positive profits.
Further, to the extent that bidders are loss averse (have strong disutility for negative profits;
Kahneman and Tversky, 1979) this will definitely reduce bids as in region 2 as bidders face the
possibility of earning a single unit at a price greater than vh. Loss aversion will also reduce bids
in part of region 3 where bidders are not assured of earning positive profits, as in going for it may
earn both units but at a price that is less than their valuation, even accounting for the synergy
bonus.
B. Ascending-bid (Clock) Auctions: The ascending-bid version of the uniform-price auction (also
referred to as a clock auction or an English-clock auction) starts with the price being zero and
increasing rapidly thereafter. Bidders start out actively bidding on all units demanded, choosing
the price to drop out of the bidding. Dropping out is irrevocable so that a bidder can no longer
11
bid on a unit she has dropped out on.7 The dropout price which equates the number of remaining
active bids to the number of items for sale sets the market price. Winning bidders earn a profit
equal to the value of their winning unit less the market price. All other units earn zero profit.
Posted on each bidder's screen at all times is the current price of the item, the number of items for
sale, and the number of units actively bid on, so that h can tell at exactly what price a rival has
dropped out. Further, there is a brief pause in the progress of the price clock following a drop out
during which h can drop out as well. Dropouts during the pause are recorded as having dropped
out at the same price, but are indexed as having dropped later than the dropout that initiated the
pause.8
Bidders i = 1, ..., n demanding a single unit have a dominant strategy to remain active until
the clock price reaches their value, vi. As in the sealed-bid version of the auction there are three
regions of interest (see the bottom panel of Figure 1):
Region 1: For vh < ½, the optimal bidding strategy for h is comparable to the sealed-bid auction
strategy in the sense that h earns greater expected profit by winning a single unit and reducing the
price paid by not winning a second unit. The strategy below assures winning at most one unit.
However, there is considerably more flexibility in carrying out the optimal policy than in the
sealed-bid auction:
If v2 # vh, d1 = vh and 0 # d2 # v2, and
If v2 > vh, then d1 0 [vh, max (vh, v3)] and d2 0 [0, max (vh, v3)]
where d1 and d2 are h’s dropout price on units 1 and 2, respectively, and v2 and v3 are,
respectively, the observed drop-out prices of the single-unit bidders with the n-1 (second) and n-2
(third)) highest valuations.
Region 2: For v 0 [½, 2/3) the optimal bidding strategy uses the information revealed by rivals’
dropout prices. In particular there is a cutoff point, P* = [3vh !1] such that:
12
If v2 # P*, d1 = d2 $ 1 and
If v2 > P*, d1 = d2 0 [P*, max {P*, v3}].
In contrast to the sealed-bid auction, the bidding rule in this region does not depend on the
number of rivals as the information revealed in the second highest computer’s dropout price is
sufficient to determine the optimal bidding strategy.
Region 3: For vh $2/3, d2 $1, so h “goes for it,” winning both units for sure. Note that “going for
it” in this interval assures positive profits, as h earns at least 2 and pays at most 2, so that the
exposure problem is not an issue here. This strategy yields higher expected profits for any
realization of v2 compared to stopping the auction and earning a single unit. Further, the size of
this interval is smaller than the interval in which bidders “go for it” in the sealed-bid auctions.
Risk aversion has no affect on bidding in region 1, since like in the sealed bid auction, b1
= vh is based on a dominance argument, and the effect of risk aversion is to reduce h’s bid on unit
2, unless b2 = 0. Risk aversion has no affect throughout region 3 as bidders are certain of winning
and earning nonnegative profits throughout region 3. But, as in the sealed-bid auctions, it does
impact on bidding in region 2. Here, bidders in going for it, earn both units but suffer losses as a
consequence. Thus, to the extent that agents are loss averse, for which there is substantial
empirical evidence in individual choice studies, this would definitely suppress the aggressive
bidding called for when v2 ≤ P* and may even get bidders to drop out before P* when vh < P*.
Clearly both auction formats will produce inefficiencies. In region 1 this is the result of
demand reduction by the multi-unit demand bidder. There will also be some ex post inefficiencies
in region 2 as h bids based on the expected values of the single-unit bidders. However, for those
valuations that fall in region 2 in the clock auctions efficiency will tend to be higher in the clock
auctions as bidders condition their actions on observed dropout prices.
Revenues will be the same under both auction formats in region 1 and in the part of region
13
3 where going for it assures nonnegative profits. For those valuations that fall in region 2 in the
clock auctions, but region 3 in the sealed-bid auctions, revenues are bound to be higher in the
sealed-bid auctions as bidders go for it in all cases, whereas in the clock auctions they drop out if
v2 ≤ P*. In the remainder of region 2 revenues will tend to be higher in the clock auctions as
conditioning on observed dropout prices often requires bidders to go for it, as opposed to bidding
somewhat above their valuation in the sealed-bid auctions.
2. Experimental Design and Theoretical Implications for Treatments Employed
A. Procedures. Valuations were i.i.d. from a uniform distribution with support [0, $7.50].9
Bidders with single-unit demands were represented by computers programmed to submit bids
equal to their value in the sealed-bid auctions or to stay active until the price reached their value
in the clock auctions. Bidder h was played by subjects drawn from a wide cross-section of
undergraduate and graduate students at the University of Pittsburgh and Carnegie-Mellon
University. Students were recruited through fliers posted throughout both campuses,
advertisements in student newspapers, electronic bulletin board postings, and classroom visits.
Each h operated in her own market with her own set of computer rivals, knew they were bidding
against computer rivals and the number of computer rivals, as well as the computers’ bidding
strategy.
The use of computer rivals has a number of advantages in a first foray into this area: hs
face all of the essential strategic tradeoffs involved in auctions of this sort but in a very “clean”
environment. The latter include no strategic uncertainty regarding other bidders’ behavior and no
issues of whether or not “common knowledge” assumptions are satisfied.
All clock auctions employ a “digital” clock with price increments of $0.01 every 0.1
second. Following each computer drop out there was a brief pause of 3 seconds. h’s dropping
out during a pause counted as dropping at the same price, but later than, the computer’s dropout.
14
h could drop out on a single unit by hitting any key other than the number 2 key. The first stroke
of the key pad dropped out unit 2. Hitting the number 2 key, or a second key during the pause,
permitted h to drop out on both units at the same price.
In the sealed-bid auctions the sequencing required subjects to submit bids on unit 1
followed by unit 2. Any nonnegative bid was accepted for unit 1, with the bid for unit 2 required
to be the same or lower than the bid on unit 1.10 There was no opportunity to submit a single bid
for both units combined or a bid contingent on winning only one or winning both units.
In all sessions, instructions were read out loud, with copies for subjects to read as well.
The instructions included examples of how the auctions worked as well as indicating some of the
basic strategic considerations inherent in the auctions. For example, the instructions pointed out
that the higher h’s value, the more valuable the synergy bonus was, hence the greater value of
earning two rather than one unit, and that when bidding above vh winning a single unit would
necessarily involve losses (see the instructions for full details). Finally, it was emphasized to
subjects that “...in thinking about bidding, earning an item is of no intrinsic value. Your sole
objective should be to maximize your earnings.”
Subjects were told that in each auction period the computers would (and did) receive fresh
values. At the conclusion of each auction, bids were ranked from highest to lowest and posted
along with the corresponding values. Winning bids were identified, prices were posted, profits
were calculated, and cash balances were updated. Bidders only observed the outcomes for their
own market. Sessions began with three dry runs to familiarize subjects with the auction
procedures, followed by thirty-three auctions played for cash.
Bidders were given starting capital balances of $5. Positive profits were added to this and
negative profits subtracted from it. End-of-experiment balances were paid in cash. Expected
profits were sufficiently high that we did not provide any participation fee.11 Sessions lasted
15
between 1.5 and 2 hours.
B. Treatments and Their Theoretical Implications. Since single-unit bidders have a dominant
strategy independent of h’s valuation, this permits us to employ a limited number of values for h
without distorting the equilibrium predictions. Given the complexity of the environment, we
limited ourselves to four values designed to fully represent/span the strategy space, and to induce
maximum differences in strategic behavior between the sealed-bid and clock auctions. By
repeating the use of the same valuations within an experimental session we provide bidders with
considerable experience at each value, which might be expected to ease decision making in such a
complex environment, while providing us with multiple observations against which to evaluate
behavior.
Sessions employed either three or five computer rivals (n = 3 or 5), with the number of
computer rivals remaining constant within each session. In each session vh varied randomly over
three of the four values employed using a block random design. All three values occurred in each
consecutive series of three auctions, but in random sequence. The lowest vh, $3.00, calls for
complete demand reduction in both auctions, and is employed exclusively with n = 3. (The
expected cost of deviating from the optimal strategy with vh = $3.00 and n = 5 is quite small, and
involves a rather large opportunity cost in terms of foregone observations at more salient values.)
The highest vh, $5.10, requires “going for it,” and insures a secure (minimum) profit of 30¢ per
auction. It is employed exclusively with n =5 out of cost considerations and the fact that there is
little ‘bang for the buck’ at this value in auctions with n = 3.12 It provides a check on the first
order rationality of bidders (and/or their sensitivity to the underlying economic contingencies).
The middle vh’ s employed make different predictions between sealed-bid and clock
auctions and were employed with both n = 3 and 5. These are clearly the most difficult values for
bidders to get right as well as the most critical values for distinguishing the extent to which they
16
are response to the underlying economic forces at work.
vh = $4.00: In the sealed-bid auctions b1 = b2 = $4.34 with n = 3 and b1 = b2 = $4.16 with n
= 5. The clock auction also requires bidding above value on both units: If v2 # P* = $4.50, h goes
for it as the expected value of winning two units is positive and greater than the value of stopping
the auction at p = v2 and winning a single unit. If v2 > P*, h drops on both units at the cutoff point
P*.
vh = $4.40: It pays to “go for it” in the sealed-bid auctions regardless of whether n = 3 or
5. The clock auction also calls for bidding above value, with the cutoff point P* = $5.70.
Finally, for both intermediate values, if h mistakenly stays beyond max{P*, v3} when v2 >
P*, the certain loss associated with stopping the auction and winning one unit is greater than the
expected loss associated with remaining active and winning both units. Thus in equilibrium, or
outside equilibrium, at these intermediate values, optimal bidding requires that h should never
win only one unit in the clock auction.
The top part of Table 1 summarizes these predictions regarding h’s bids for each of the
computer values employed. While these predictions are of primary interest, there are also clear
implications for h’s profits, seller revenue, and auction efficiency. Profits, revenue, and
efficiency all predicted to be the same between auction formats within regions 1 and 3 (vh = $3.00
and $5.10), as bid outcomes are predicted to be the same in these regions. Profits, revenue, and
efficiency are all predicted to be substantially higher in the clock auction with vh = $4.00, as h
decides to go for it contingent on the observed drop-out prices of the single-unit bidders. This
makes for much finer distinctions than under h’s relatively conservative, fixed bidding strategy in
the sealed-bid auctions. Note that profits and revenue can both increase in this treatment due to
the increased efficiency. At vh = $4.40, revenue is predicted to be higher in the sealed-bid
auctions as bidders go for it, while in the clock auctions going for it is contingent on the observed
17
drop-out prices of the single-unit bidders. As a consequence profits are predicted to be higher
under the clock auctions. However, there is very little to choose between the two auction formats
on efficiency grounds. The intuition for this last result is that ex post, at times bidders should
have gone for it in the clock auction, whereas they always go for it in the sealed bid auctions and
these two effects are essentially offsetting for vh = $4.40. These predictions regarding revenue,
profits and efficiency are summarized in the bottom half of Table 1.
[Insert Table 1 around here]
3. Experimental Results
Our focus will be on bidding behavior, with some attention to revenue and efficiency as
well. Throughout the analysis we will concentrate on bidding in the last 6 auctions under each vh,
when bidders would have become reasonably familiar with the environment (recall, that there
were 11 auctions played for cash at each vh, and 1 dry run).13
A. Bid Patterns: Table 2 compares the frequency of optimal play between the two auction
institutions. This table employs rather strict definitions for optimal play, with the exception of
providing 5¢ “allowances” for “trembles” throughout. ( For example, with vh = $3.00, in the clock
auctions we count as equilibrium d2 # v2 + 0.05 when v2 # vh. Our results are robust to either
eliminating these allowances or increasing them a bit.) Conclusion 1 is based on these results.
[Insert Table 2 around here]
Conclusion 1: Bids are closer to optimal outcomes in the clock auction for all valuations.
Bids being closer to equilibrium/optimal play in clock versus sealed-bid auctions are
consistent with results from a large number of auction experiments: single-unit private value
auctions (Kagel, Harstad, and Levin, 1987), single-unit common value auctions (Levin, Kagel,
and Richard, 1996), and multi-unit demand, uniform-price auctions without synergies (Kagel and
Levin, 2001; also see Kagel (1995) and Kagel and Levin (2002) for reviews of these results.) The
18
experimental manipulations reported in Kagel and Levin (2001) suggest that it is the clock
auction format, in conjunction with the information revealed to h by dropout prices, that is
responsible for its superior performance.14
What’s missing from Table 2, and will provide the focus of the detailed analysis that
follows, is the pattern of deviations from equilibrium, which is quite different between the two
auction institutions.
Conclusion 2: In the sealed-bid auctions the data show a clear pattern of increasing bids at
higher values as the theory predicts. However, bidders are imperfectly calibrated, as bids are
substantially higher than they should be at lower values, and are lower than they should be at
higher values, with the possible exception of vh = $5.10, where bids are close to optimal.
[Insert Table 3 around here]
Support for this conclusion is reported in Table 3, where we have fit random effect Tobits
to the bid data. We employ Tobits as there is a mass point at $7.50, and we truncate all bids
greater than $7.50. An error components specification is employed with the error term ,it = *i +
.it, where *i is a subject-specific error term assumed to be constant between auctions within an
experimental session, and .it is an auction period error term. Standard assumptions regarding the
error terms are employed; i.e., *i -(0, F*) and .it -(0, F.) where *i and .it are independent among
each other and among themselves. (Note, in our experimental design, the use of computer rivals,
clearly supports the assumption that the *i are independent among each other.) With n = 3 we use
vh = $3.00 as the intercept of the bid function and create a dummy variable (DV4) for vh = $4.00
(DV4 = 1 if vh $ $4.00, 0 otherwise), and a second dummy, (DV440) for vh = $4.40 (DV440 = 1 if
vh = $4.40, 0 otherwise). For n = 5 we use vh = $4.00 for the intercept of the bid function and
create separate dummy variables vh = $4.40 (DV440 = 1 if vh $ $4.40, 0 otherwise) and for vh =
$5.10 (DV510 = 1 if vh = $5.10, 0 otherwise). Also reported, for the reader’s convenience, are the
19
95% confidence intervals for bids predicted by the regression equations.
For the n = 3 both dummy variables are significant at the 5% level or better, indicating
that bids were increasing in vh, as they should be, for both b1 and b2. However, bids are
substantially higher than they should be at lower values, and are lower than they should be at
higher values, with the possible exception of vh = $5.10, where bids are close to optimal. For vh =
$3.00 and $4.00 the lower bound of the 95% confidence interval for unit 1 bids is well above the
optimal bid, with unit 2 bids showing a similar pattern. For vh = $4.40 the upper bound of the 95%
confidence interval for unit 1 bids is $7.50, as it should be, the upper bound for unit 2 bids falls
well short of the optimal bid of $7.50. Finally, for vh = $4.00 the upper bound for unit 2 bids falls
short of the lower bound for unit 1 bids. This is indicative of a relatively high frequency with
which b1 ≠ b2 when bidding above value, in violation of dominance arguments (but a dominance
argument that would be far from transparent to most subjects).15
Similar results are reported for the n = 5 case. Both dummy variables are significant at the
5% level or better, indicating that bids were increasing in vh, as they should be, for both b1 and b2.
For vh = $4.00, the lower bound of the 95% confidence interval for unit 1 bids is well above the
optimal bid of $4.16, as is the lower bound for unit 2 bids. For vh = $4.40, the upper bound of the
95% confidence interval for unit 1 bids includes $7.50, but is short of the optimal bid of $7.50 for
unit 2 bids. What is different from the n = 3 case is that there is overlap between unit 1 and unit 2
bids in all cases, indicating that we cannot reject a null hypothesis that b1 = b2, at least on the
aggregate level.16 No doubt these differences result from the fact that bidding less on unit 2
compared to unit 1 (conditional on b1 > vh) is more costly with n = 5 than n = 3.17 Finally, the
regression shows that for vh = $5.10, the lower bound of the 95% confidence interval for unit 1
bids is $7.50, and the 95% confidence interval for unit 2 bids includes the optimal bid of $7.50 as
well. As such, even though the equilibrium requirement of b1 $ b2 $$7.50 is only satisfied 40.6%
20
of the time, bidders win both units 71.9% of the time. The net result is that deviations from
optimality in payoff space are substantially smaller than deviations from optimality in the choice
space.
Conclusion 3: There is directional consistency in the clock auctions in the sense that two units
are won more often at higher valuations (when they should be won) than at lower valuations
(when they should not win two units). Further, at vh = $5.10 bidders “go for it” most of the time,
as they should. However, at the intermediate valuations of region 2, there are large deviations
from optimal bid patterns that are best explained by the exposure problem.
Support for Conclusion 3 can be found in Table 4. First note that for v2 # vh the frequency
of winning two units is increasing in vh. Further, consistent with optimal bidding, two units are
won close to 100% of the time (in 83.3% of all auctions) with vh = $5.10. However, substantial
deviations from optimal bid patterns are reported in region 2, at the intermediate values of $4.00
[Insert Table 4 around here]
and $4.40, where bidders face an exposure problem. This exposure problem expresses itself in
three distinct ways for these two valuations:
(1) For v2 # vh the primary deviation from equilibrium bidding involves demand reduction
(winning 1 unit with positive profits). Further, in most cases this involves complete demand
reduction; i.e., dropping out at the same time (or prior to) v2, thereby not affecting the market
price (87.5% of all cases with vh = $4.00, 74.7% of all cases with vh = $4.40) and never having to
bid above vh.
(2) In cases where vh < v2 # P*, where optimality requires “going for it,” and necessarily
involves bidding above vh, there is a high frequency of winning zero units. The combined
frequency of winning zero units and winning one unit (which involves “going for it” but then
reconsidering when bidding above vh) is consistently higher than the frequency of winning two
21
units. Further, random effect probits show that bidders win two units significantly less often than
when v2 # vh. In contrast, the theory predicts no difference in the frequency of winning two units
between these two cases.
(3) In cases where v2 > max(P*, vh) equilibrium play calls for bidding up to P* and
dropping on both units at this point. This necessarily involves bidding above vh, but rarely
happens. Rather the modal response in three out of four of these cases is to drop out prior to P*,
typically dropping out very close to vh (and dropping too soon is within a whisker of the modal
response for the fourth case - n = 3 and vh = $4.40).
In contrast to bidding at these intermediate valuations ($4.00 and $4.40), bidders have
little problem bidding above their valuation for vh = $5.10, when they are assured of a minimum
profit of 30¢ by going for it. This response to the exposure problem in the clock auction is in
marked contrast to the sealed-bid auction where bidders show no reluctance to bid above vh on
both units. This suggests that it is both the fear of losses, in conjunction with the auction format,
that is responsible for the greater response to the exposure problem in the clock auction. What is
it about the clock auction that accounts for this heightened fear of losses? In single-unit private
value auctions when it is a dominant strategy to bid your valuation, there is close conformity to
the dominant bidding strategy in clock auctions (with feedback on bidders’ dropout prices), as
opposed to substantial overbidding in sealed-bid auctions (Kagel et al., 1986). The evidence
suggests that this results from the fact that it is much more transparent to bidders in the clock
auctions that losses can and do occur strictly as a consequence of bidding above value (Kagel et
al., 1986; Kagel, 1995; Kagel and Levin, 2001). In single-unit private value auctions, and in
multi-unit demand auctions without synergies, this heightened awareness of the perils of bidding
above one’s valuation helps to improve bidder profits and to move play closer to the equilibrium
outcome. Here it holds bidders back from achieving maximum profit and promotes deviations
22
from the equilibrium outcome.
One final thing to note in Table 4 is the sharp (and consistent) difference between the
frequency of winning two units for n = 3 versus n = 5 when v2 # vn, and when v2 # P* for the
region 2 valuations of $4.00 and $4.40.18 We suspect that this has little to do with the different
number of rivals in the two treatments, but is a hysteresis effect brought on by the different
behavior patterns rewarded under the remaining valuation in each case: vh = $3.00 calls for
complete demand reduction with n = 3 and vh = $5.10 calls for “going for it” with n = 5. This
result is summarized in Conclusion 4.
Conclusion 4: There appears to be a strong hysteresis effect in the data, as with resale values of
$4.00 and $4.40, the exposure problem is much more severe with n = 3 than with n = 5.
The next conclusion takes a closer look at bidding with vh = $5.10. In almost all respects
this should be (and is) the valuation for which play is closest to optimal in both auction formats as
it only takes a little arithmetic to realize that “going for it” yield a secure, minimum profit of 30¢
per auction. As a result, with repeated exposure to the problem one would expect more subjects to
“get it.” And they do, as the data show a clear learning effect, converging toward optimal play.
Conclusion 5: With vh = $5.10, we observe clear adjustments over time toward optimal play in
both sealed-bid and clock auctions. However, winning two units is still more pronounced in the
clock auctions.
Support is provided by the random effect probits reported in Table 5, where we have
pooled the data for both clock and sealed-bid auctions for vh = $5.10. The dependent variable
takes on a value of 1 in cases where two units were won (as optimal bidding requires) and 0
otherwise. We use the data for all auctions, excluding the dry runs. Model 1 includes a single
dummy variable, Dclock = 1 if a clock auction, 0 if sealed bid. The coefficient for the dummy is
positive and significant at the 10% level, indicating that play is closer to the optimal outcome in
23
the clock auctions. Model 2 introduces a second dummy variable, Dearly = 1 for the first 5
auctions with vh = $5.10, 0 for the last 6, in an effort to identify possible learning/adjustment
effects. These are clearly present as indicated by the relatively large, statistically significant
negative coefficient value for the Dearly dummy. Finally, the introduction of the Dearly dummy
has virtually no effect on the magnitude of the Dclock dummy or its standard error.19
[Insert Table 5 around here]
This is the one case where we observe clear learning/adjustments toward optimal play in
the data. Specifications searching for learning/adjustment effects for other valuations reveal
considerably more noise in early play (higher variances and less stable coefficient values), as
opposed to any clear adjustments toward optimal play.20
B. Profits, Efficiency and Revenue: Bidders’ profits provide a measure of success in terms of
payoffs, and provide a convenient way of characterizing performance in terms of a single
outcome measure. Employing the standard measure of efficiency for auctions of this sort (see
Ledyard et al., 1997 or Plott, 1997), efficiency is defined as the sum of the values of two units
sold in each auction period (including the synergy bonus, if relevant) as a percentage of the
highest total value for two units. Note that 100% efficiency is not always attainable at the Nash
equilibrium for our game. Revenue is what the seller would have earned in each period. In
computing revenue and efficiency we are keenly aware that the same results might not emerge in
auctions where all bidders are human. As noted, computer rivals were employed to minimize
possible complications associated with learning against human rivals who may be playing out-ofequilibrium strategies. Out-of-equilibrium play may affect different institutions differently. (See,
for example, Katok and Roth, 2004, who demonstrate that the ascending bid auction is much
more sensitive to out-of-equilibrium play by small bidders than the descending bid auction.) On
the other hand, there is very limited experimental data on efficiency and revenue, measures of
24
central importance to economists, in environments such as this, and we believe that the present
data are at least suggestive of what will be observed in interactive settings. Finally, in reporting
revenue and efficiency we provide a benchmark against which future experiments with all human
bidders can compare results on these important issues.
Conclusion 6: Profits are consistently and significantly less than would have been achieved with
optimal bidding in all but one case (vh = $4.00, n = 3, sealed-bid auction). Realized profits are
always higher in the clock auctions, but in a number of cases the significantly higher profits the
theory predicts fail to be realized.
Table 6 reports profits -- actual, predicted and the difference between the two -- for both
auctions, along with the difference between actual profits in the two auctions (sealed-bid less
clock). At vh = $3.00, bidders earned negative profits averaging -60¢ and -15¢ in the sealed-bid
and clock auctions respectively. Using subject averages as the unit of observation, profits in the
sealed-bid auctions were significantly below zero (t = 2.80, p < .01, 2-tailed test), and
significantly less than in the clock auctions (t = -1.93, p < .10, 2-tailed test), reflective of the large
differences in equilibrium play between the two auctions. For n = 3, with vh = $4.00 and $4.40
optimal bidding predicts 20–24% higher profits in clock compared to sealed-bid auctions. These
higher predicted profits result from the greater flexibility afforded by the information revelation
in the clock auctions. However, these better profit opportunities do not result in significantly
higher realized profits as (i) the exposure problem serves to promote the strong demand reduction
found in the clock auctions, which wipes out most of the advantages resulting from information
revelation in this case, and (ii) the deviations from equilibrium in the sealed-bid auctions are not
severe enough to impact significantly on earnings. With n = 5, there are even larger percentage
differences between predicted profits in clock versus sealed-bid auctions for vh = $4.00 and $4.40.
These are realized in one case (vh = $4.00), with reasonably large, but statistically insignificant,
25
differences in the predicted direction in the other
[Insert Table 6 around here]
case (vh = $4.40). The theory comes off better here compared to n = 3 case because (i) the demand
reduction effect is weaker in the clock auctions with n = 5 (recall Conclusion 4) and (ii) there are
larger reductions in realized profits compared to predicted profits in the sealed-bid auctions with
n = 5. Finally, what differences there are in bidding with vh = $5.10 do not translate into
significant differences in earnings between the two auction formats (t = -1.29).21
Conclusion 7: Optimal bidding predicts either the same or higher efficiency in clock compared to
sealed-bid auctions. In contrast to these predictions, actual efficiency differences are quite mixed,
with the only significant difference recorded in favor of the sealed-bid auction.
Table 7 reports efficiency outcomes. At the $3.00 value actual efficiency is very close to
predicted levels in both auctions. This is not surprising for the clock auctions where bidding is
relatively close to equilibrium, but is somewhat unexpected in the sealed-bid auctions with its
large deviations from equilibrium outcomes. These deviations apparently have minimal impact on
efficiency since the overbidding occasionally produces large efficiency gains (relative to the
predicted outcome) as a result of the synergy bonus. At the $4.00 value, the clock auction should
yield large efficiency gains compared to the sealed-bid auction. However, these gains go almost
entirely unrealized as (i) the exposure problem serves to promote demand reduction in the clock
auctions, which wipes out most of the efficiency gains predicted, and (ii) deviations from
equilibrium bidding in the sealed-bid auctions have essentially no effect relative to predicted
efficiency. This last result is due to the fact that with the synergy bonus overbidding occasionally
produces large efficiency gains relative to the predicted outcome.22 Finally, actual efficiency is
well below predicted efficiency at higher values in both auctions as bidders do not “go for it”
often enough to take full advantage of the synergy bonus.
26
[Insert Table 7 around here]
Conclusion 8: Revenue is consistently higher in the sealed-bid auctions, and is significantly
higher in four out of six cases. The higher revenues do not come at the expense of any significant
efficiency loss relative to the clock auctions.
Revenues are reported in Table 8. Revenues are predicted to be the same at vh = $3.00 and
$5.10, to be higher in the clock auctions with vh = $4.00, and to be higher in the sealed-bid
auctions with vh = $4.40. The actual data, however, show uniformly higher revenues in the sealedbid auction: 4.0% to 18.8% higher with n = 3, 3.0% to 12.7% higher with n = 5 (calculated as a
percentage of realized revenue in the clock auctions). Note that these increased revenues do not
come at the expense of reduced efficiencies, as Table 7 reports no significant decreases in
efficiency in sealed-bid compared to clock auctions, and one case with significantly higher
efficiency in the sealed-bid auction.
[Insert Table 8 around here]
4. Summary and Conclusions
We develop a model of multi-unit demand auctions with synergies, and explore behavior
experimentally, comparing sealed-bid and ascending-bid uniform-price auctions (winning bidders
pay the highest rejected bid). We employ a simple, tractable demand structure: Several singleunit demand bidders and one bidder demanding up to two units. We further simplify the structure
by having computers play the dominant strategy that single-unit bidders have of always bidding
their value. In spite of its simplicity, the key economic incentives present in uniform-price
auctions with synergies are all captured: the synergy effect, which promotes bidding above
standalone values; the exposure problem which may deter bidders from pursing this aggressive
biding strategy, thereby reducing economic efficiency; and the monopsony power that multi-unit
demand bidders can exploit to reduce prices in a uniform-price auction.
27
The experiment shows that bidding is closer to optimal outcomes in the clock auctions,
consistent with evidence from a number of other auction environments (Kagel, et al., 1987; Levin,
et al., 1996; Kagel and Levin, 2001). Further, although they do not bid optimally, in most cases
bidders behave sensibly: Bidding under the highest valuation, where the optimal play is most
transparent, generates levels of optimal play that are comparable to the highest levels reported for
subjects in any auction experiment. Demand functions estimated for the sealed-bid auctions are
monotonically increasing in bidders’ valuations. In the clock auctions, there is a higher frequency
of “going for it” at higher valuations, when multi-unit demand bidders should attempt to obtain
both units.
Nevertheless, there is considerable out-of-equilibrium play under both institutions, with
systematic differences in the pattern of out-of-equilibrium play. The most interesting and
dramatic differences occur at intermediate valuations where the theory requires balancing demand
reduction incentives against the synergy bonus, while exposing bidders to possible losses. At
these values the exposure problem promotes relatively strong demand reduction in the clock
auctions, whereas optimal bidding requires that bidders “go for it.” In contrast, there is barely
any response to the exposure problem in the sealed-bid auctions, with bidders consistently
bidding above value on both units. This suggests that it is both the fact that bidders are exposed to
losses, in conjunction with the auction format, that is responsible for the greater response to the
exposure problem in the clock auction. The clock auction format (with feedback on bidders
dropout prices) makes it much more transparent to bidders that losses can and do occur strictly as
a consequence of bidding above value (Kagel et al., 1987; Kagel, 1995; Kagel and Levin, 2001).
This heightened awareness of the perils of bidding above ones’ valuation helps to improve bidder
profits and to move play closer to the equilibrium outcome in single-unit private value auctions
and in multi-unit demand auctions without synergies. With synergies it holds bidders back from
28
achieving maximum profit and generates deviations from the equilibrium outcome.23
The element of the exposure problem at play in our experiment stems from loss aversion.24
One might argue that this is strictly a psychological phenomena as fully rational bidders should
not be deterred by the prospect of losses from bidding aggressively to maximize expected profits.
Be that as it may, this does not make the problem any less real.
How relevant are these findings of out-of-equilibrium play to “real world” auctions? We
argue that for low stakes field settings, or those that for one reason or another, bidders do not
employ high-powered consultants (as occurred in at least one U. S. spectrum auction), our
observations are directly relevant. (See, for example Naik (1996) who discusses problems small
firms had in PCS auctions, and Mills (1997) who discusses problems a number of bidders had in
the Interactive Video and Data Services (IVDS) auction.)
But what about high-stakes settings with high-powered consultants on all sides, as is
typically the case in auctions involving many millions or even billions of dollars? Extrapolating
laboratory results to such cases is clearly more speculative and one must be much more careful in
assessing the applicability of the results. However, we do know that the exposure problem has
been a major concern in designing auctions with synergies. What our experiment highlights is that
the problem is likely to be more prevalent in an ascending price format than in a sealed-bid
auction format.
There are a number of obvious and interesting extensions to the experimental results
reported here. One would be to conduct these auctions with all human bidders to see what
differences possible out-of-equilibrium play by single-unit bidders would have on multi-unit
demand bidders. Another would be to permit the use of package bidding, either under the present
set-up or with all human bidders, to see how well this serves to overcome the exposure problem
and to identify what, if any, “complexity problems” this might pose for bidders. In addition, a
29
natural next step would be to extend the analysis to environments with several multi-unit demand
bidders and/or to environments in which items are imperfect substitutes and sold in separate
auction markets. And to consider some of the other tradeoffs between sealed-bid and clock
auctions such as the increased costs of clock auctions (both in terms of configuring bidders and the
longer time required to conduct such auctions), the clock’s potential for information aggregation in
common value auctions, and differences in collusive possibilities between the two auction
formats.25
30
1. Particular attention has been given to the effects of uniform-price auction rules because they are relatively easy to
characterize and to implement, and are reasonably close in format to the one employed in the spectrum auctions (see
Cramton, 1995).
2. Bykowsky et al. discuss two types of exposure problems that may exist in complex environments with synergies.
In our simple environment only the first of these potential problems exists, namely exposure to bidding above the
stand-alone value and not obtaining the desired package, or obtaining it but at higher prices than anticipated. Bidder
responsiveness to the exposure problem in this case is akin to loss aversion (Kahneman and Tversky, 1979).
Package bidding has its own problems. These include the free rider/threshold problem and the computational
complexity problem (Charles River Associates and Market Design Inc., 1998).
3.That is, we announce the fact that subjects are playing against computers and that the computers will always bid
their value, but do not discuss the basis for the computers’ bidding strategy.
4.Krishna and Rosenthal extend their analysis to auctions in which there is more than one bidder with increasing
returns. The Krishna-Rosenthal model is (arguably) closer, in some respects, to the U. S. spectrum auctions than our
experimental design. Their model does not, however, deal with possible synergies within a given market as ours
does. It is these underlying economic and behavioral forces that our experiment is designed to investigate rather than
any effort to faithfully replicate any particular spectrum sale design.
5.This result comes about for different reasons in the two models: In our case, bidding the same on both items
follows from a dominance argument (see Appendix A). In Krishna and Rosenthal it follows from the assumption of
equal numbers of single-unit bidders in each market in conjunction with their focusing on symmetric Nash bidding
strategies.
6.We are keenly aware that the way synergies have been modeled here is not without loss of generality. However,
we need to implement a design that is both theoretically tractable and easy to implement and explain. Setting " = 1
and m = 2 greatly simplifies bidders’ calculations while still yielding the rich behavioral space characterized in
Figure 1 below.
7.Given that the computers follow the dominant strategy, h has no incentive to drop out and re-enter the auction.
However, we plan to conduct additional experiments where the irrevocable exit rule may become relevant.
31
8.The auction is formally modeled as a continuous-time game. However, we want to take into account the
possibility that bidder i’s strategy may be to reduce her quantity at a given time, while bidder j’s strategy may be to
reduce his quantity at the soonest possible instant after bidder i does. This requires allowing “moves that occur
consecutively at the same moment in time” (Simon and Stinchcombe, 1989; also see Ausubel, 1997).
9.The underlying support for valuations, along with the number of computer rivals, were chosen with an eye towards
comparing behavior here with earlier multi-unit demand auctions with flat demands (no synergies) which call for
demand reduction at all values (Kagel and Levin, 2001).
10.This restriction was built into the software. We have tested for the impact of this restriction in other multi-unit
demand auctions (without synergies), with these tests showing that this restriction has no effect on bidding strategies
(Kagel and Levin, 2001).
11.In those few cases where end-of-experiment earnings were below $2.00, a token $2.00 payment was provided.
This was not announced in advance and only applied to two or three subjects.
12.In the clock auctions with n = 3, v1 is often below $5.10, so there is very little information to be gained about
whether or not h’s recognize that “going for it” is the optimal strategy in these cases. This problem is reduced
substantially with n = 5. Further, rather substantial deviations from the optimal bidding strategy of b1 = b2 $$7.50
(for example, bidding above $5.10 but below $7.50) frequently incur no penalty in sealed-bid auctions with n = 3,
but such deviations are much more likely to be punished with n = 5.
13.The choice of the last 6 auctions is somewhat arbitrary, but it does distinguish more from less experienced play,
and the results are robust to adding or dropping an auction or two to either side of 6. We will occasionally make
reference to obvious learning/adjustment patterns in bids, but forgo any kind of detailed analysis as the paper is
already quite long.
14.Kagel and Levin (2001) show that in a multi-unit demand, uniform-price auction, with flat demand (i.e., with no
synergies so that only the demand reduction effect is in present), a clock auction with no feedback on rivals’ drop-out
prices looks no different from the sealed-bid version of the auction. Further, a sealed-bid auction with the crucial
dropout information employed in the clock auction provided by the experimenter, improves performance, but still
comes up short compared to a clock auction with rivals’ drop-out prices announced.
32
15.The raw data on this score is as follows: Conditional on b1 > vh + 0.05, the frequency of b1 - b2 > .25 or b2 # vh is
51.0% with vh = $3.00, 35.6% with vh = $4.00 and 31.9% with vh = $4.40. (Note, our calculations provide small
allowances for deviations from optimal play to leave some room for “trembles.”)
16.This is clear from the raw data as well, which shows that conditional on b1 $ vh, the relative frequency of b1 > b2
has been cut by 50% compared to n =3: for n = 5, conditional on b1 > vh + 0.05, the frequency of b1 - b2 > .25 or b2 #
vh is 17.2% with vh = $4.00, 17.7% with vh = $4.40 and 12.0% with vh = $5.10.
17.For example, with vh = $4.00, numerical evaluation of outcomes for the rule of thumb, b1 = 1.5vh and b2 = vh,
yields positive profits, but profits that are lower by approximately 33¢ per auction, than the optimal strategy with n
= 3. With n = 5, this rule yields negative profits which are lower by approximately 53¢ per auction than profits
generated by the optimal strategy.
18.These differences are statistically significant at conventional levels. Tests for statistical significance consisted of
the following: Take all cases where v2 # vh. Run a random effects probit (with subject as the random component) and
dependent variable = 1 when a bidder wins 1 unit with positive profits), 0 otherwise. Let vh = $3.00 serve as the
baseline and define dummy variables DV4 = 1 if vh $ $4.00, 0 otherwise; DV440 = 1 if vh $ $4.40, 0 otherwise; and
DV510 = 1 if vh = $5.10, 0 otherwise; and DN5 = 1 if n = 5, 0 otherwise. This yields:
Win1 = 0.502 - 0.601DV4 - 0.187DV440 - 0.630DV510 - 0.932DN5
(0.292)+ (0.222)**
(0.159)
(0.258)**
(0.358)**
19.A third specification introducing an interaction effect between the Dearly and Dclock dummies fails to reduce
the log likelihood function by a significant amount.
20.We have made limited inquires into the effect of bringing back experienced bidders. These sessions show that
experienced subjects in sealed-bid auctions do not perform materially better than inexperienced ones. There are,
however, significant differences depending on bidders’ past experience: Those with clock experience bid less
aggressively at lower valuations compared to those with sealed-bid experience. These differences constitute a direct
carryover of the differences observed between auctions for inexperienced bidders (see www.econ.ohiostate.edu/kagel/final.pdf).
33
21.The differences we do find are aided by the higher predicted profits in the clock auctions. The latter results from
sampling variability in terms of the computer values drawn, as optimal bidding yields the same equilibrium outcome
in both cases.
22.Actual efficiency is greater than predicted efficiency in the sealed-bid auctions in these two cases. There is no
inconsistency here since predicted efficiency is less than 100%.
23.Katok and Roth (2004) compare descending price “Dutch” auctions and ascending price “English” in an
environment similar to ours (homogeneous goods sold in a single market with one large and several small bidders).
One of their treatment conditions in the ascending price auctions presents an exposure problem for the large bidder
in that the equilibrium calls for bidding above his/her stand alone value for a single unit. In 38% (49/129) of all such
auctions the large bidder is not willing to bid above this stand alone value, which represents an exposure problem
similar to the one reported here.
24.In auctions with synergies one can readily establish distributions of values for which, absent package bidding,
the only auction outcome possibilities are inefficiency (as the synergies fail to be realized) or individual losses (see,
for example, Bykowsky, et al., 1995). This too is referred to as the exposure problem. However, there is a dramatic
difference between this exposure problem, where rational bidders’ bid passively in equilibrium, and the exposure
problem identified here which is based on the behavioral phenomena of loss aversion in conjunction with the clock
auction format.
25.Alternatives to the clock auction, with much of the same potential benefits and fewer drawbacks (in theory at
least) on these dimensions are “survivor” auctions (Fujishima et al., 1999) or two-stage sealed-bid auctions (Perry et
al., 2000). We are currently conducting experiments comparing survivor auctions with sealed-bid and clock
auctions.
34
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Econometrica, 57, 1171-1214.
Table 1
Summary of Theoretical Predictions for Multi-Unit Bidder Values Employed
Equilibrium Bids
Bidder value ( vh )
Sealed Bid Auctions
Clock Auctions
$3.00
(with 3 computer rivals only)
b1 = ν h ; b2 = 0
If ν 2 < ν h , d1 = $3.00 and d 2 ≤ ν 2 .
If ν 2 > ν h , d1 ∈ [$3.00, max ($3.00, ν 3 )] and
d 2 ∈ [0, max ($3.00, ν 3 )].
$4.00
(with 3 and 5 computer rivals)
$4.40
(with 3 and 5 computer rivals)
$5.10
(with 5 computer rivals only)
with 3 computers: b1 = b2 = $4.34
If ν 2 ≤ = $4.50, d1 = d 2 = $7.50
with 5 computers: b1 = b2 = $4.16
If v2 > = $4.50, d1 = d 2 ∈ [$4.50, max ($4.50, v3 )].
b1 = b2 > $7.50
If ν 2 ≤ $5.70, d1 = d 2 = $7.50.
(earn two units)
If v2 > $5.70, d1 = d 2 ∈ [$5.70, max ($5.70, v3 )]
b1 = b2 > $7.50
d1 = d 2 = $7.50.
(earn two units)
Revenue, Profits and Efficiency
$3.00 & $5.10
Revenue, profits and efficiency are the same in both auction formats.
$4.00
Revenue, profits and efficiency are higher in the clock auctions than in the sealed bid auctions
$4.40
Revenue is higher and profits are lower in the sealed-bid auctions. Efficiency is essentially the same in the two
auctions.
Table 2
Comparing Frequency of Equilibrium Play Under Different Auction Institutions
(raw data in parentheses)
vh
No. Computers
Clock
Sealed Bid*
$3.00
3
46.3%
(111/240)
23.7%
(57/240)
22.3%
(54/240)
38.8%
(93/240)
35.8%
(86/240)
79.2%
(190/240)
2.6%
(5/189)
1.6%
(3/188)
3.1%
(6/192)
27.7%
(52/188)
27.1%
(52/192)
40.6%
(78/192)
3
$4.00
5
3
$4.40
5
$5.10
5
* For n = 3, one subject left before her session ended resulting in fewer than 6 auctions at each vh value for
that subject.
Table 3
Sealed Bid Auctions: Random Effect Tobit Estimates of Bid Function
No. of Computers
95% confidence interval for bids
[predicted bids in brackets]
b 1 = 6.19 V3 + 2.08 DV4 + 1.62 DV440
(0.782)** (0.948)* (0.989)
3
5
vh =$3.00
4.65-7.50
[3.00]
No. Subjects
No. Observations
vh =$4.00 vh =$4.40
Vh =$5.10
6.68-7.50
7.50
NA
[4.34]
[7.50]
3.23-5.16
[0.0]
4.54-6.50
[4.34]
5.20-7.18
[7.50]
NA
32
565
b 1 = 6.99 V4 + 1.04 DV440 + 1.69 DV510
(0.515)** (0.452)*
(0.482)**
NA
5.98-7.50
[4.16]
6.99-7.50
[7.50]
7.50
[7.50]
32
576
b 2 = 6.09 V4 + 0.70 DV440 + 0.88 DV510
(0.155)** (0.154)**
(0.160)**
NA
5.78-6.39
[4.16]
6.48-7.10
[7.50]
7.34-7.50
[7.50]
b 2 = 4.19V3 + 1.33 DV4 + 0.67 DV440
(0.491)** (0.513)** (0.524)
* significantly different from 0 at the .05 level, 2-tailed test
** significantly different from 0 at the .01 level, 2-tailed test
TABLE 4
Bid Patterns in Clock Auctions
(raw data in parentheses)
ν2 ≤ νh
No.
Computers
Value
$3.00
$4.00
3
$4.40
$4.00
5
$4.40
$5.10
Win 2
Units
23.7%
(18/76)
33.8%
(44/130)
47.0%
(70/149)
Demand
Reduction
63.2%
(48/76)
49.2%
(64/130)
43.0%
(64/149)
45.2%
(28/62)
60.8%
(45/74)
83.3%
(95/114)
25.8%
(16/62)
25.7%
(19/74)
11.4%
(13/114)
v2>Max{vh,P+}
P* ≥ ν 2 > ν h
Other
13.2%
(10/76)
16.9%
(22/130)
10.1%
(15/149)
Win 2
Units
NA
Win 0
Units
NA
Win 1
Unit
NA
25.0%
(6/24)
38.0%
(19/50)
50.0%
(12/24)
38.0%
(19/50)
25.0%
(6/24)
24.0%
(12/50)
29.0%
(18/62)
13.5%
(10/74)
5.3%
(6/114)
39.1%
(9/23)
47.2%
(34/72)
75.4%
(95/126)
39.1%
(9/23)
31.9%
(23/72)
14.3%
(18/126)
21.7%
(5/23)
20.8%
(15/72)
10.3%
(13/126)
Equilibrium
Drop Too
Late & Win 0
14.6%
(24/164)
8.1%
(7/86)
4.9%
(2/41)
Win 1
Win 2
44.5%
(73/164)
8.1%
(7/86)
9.8%
(4/41)
Drop Too
Soon & Win 0
13.4%
(22/164)
53.5%
(46/86)
39.0%
(16/41)
20.1%
(33/164)
17.4%
(15/86)
2.4%
(1/41)
7.3%
(12/164)
12.2%
(11/86)
43.9%
(18/41)
11.0%
(17/155)
7.4%
(7/94)
NA
47.7%
(74/155)
38.3%
(36/94)
NA
14.8%
(23/155)
12.8%
(12/94)
NA
12.9%
(20/155)
19.1%
(18/94)
NA
13.5%
(21/155)
22.3%
(21/94)
NA
Cells with outcomes in bold constitute equilibrium predictions; i.e., they should be 100% in all cases.
NA: Not applicable
Table 5
Probits Comparing Winning 2 Units in Clock vs Sealed Bid Auctions: vh = $5.10
Variable
Constant
Model 1
DEarly
0.706
(0.211)**
0.439
(0.252)+
-----
Log Likelihood
No. Observations
No. Subjects
-362.9
792
72
DClock
+ Significantly different from 0 at the 10% level, 2-tailed test
** Significantly different from 0 at the 1% level, 2-tailed test
Model 2
0.981
(0.222)**
0.458
(0.255)+
-0.540
(0.122)**
-352.8
792
72
Table 6
Profits (in dollars)
(standard errors of mean in parentheses)
No. Computers
vh value
$3.00
3
$4.00
$4.40
$4.00
5
$4.40
$5.10
Sealed Bid Auctions
Actual
Predicted Difference
Clock Auctions
Actual Predicted Difference
-0.60
(0.214)
0.72
(0.176)
1.24
(0.207)
-0.25
(0.123)
0.25
(0.144)
2.03
(0.263)
-0.15
(0.127)
0.73
(0.120)
1.29
(0.150)
0.18
(0.093)
0.46
(0.145)
2.45
(0.202)
0.35
(0.056)
0.88
(0.176)
1.74
(0.240)
0.21
(0.073)
0.54
(0.260)
2.69
(0.149)
+ Significantly different from 0 at the 10% level, 2 tailed t-test
** Significantly different from 0 at the 1% level, 2 tailed t-test
-0.959**
(0.187)
-0.168
(0.111)
-0.497**
(1.33)
-0.459**
(0.092)
-0.286**
(0.077)
-0.661**
(0162)
0.36
(0.032)
1.09
(0.144)
2.10
(0.183)
0.44
(0.104)
0.84
(0.140)
2.93
(0.138)
-0.495**
(0.124)
-0.365**
(0.104)
-0.805**
(0.146)
-0.260**
(0.074)
-0.385**
(0.074)
-0.490**
(0.129)
Difference (actual):
Sealed Bid less Clock
(t-statistics)
-0.45
(-1.93)+
-0.01
(-0.02)
-0.05
(-0.20)
-0.43
(-2.80)**
-0.21
(-0.89)
-0.42
(-1.29)
TABLE 7
EFFICIENCY
(standard errors in parentheses)
No.
Computers
vh
Value
$3.00
3
$4.00
$4.40
$4.00
5
$4.40
$5.10
Sealed Bid Auctions
Difference
(actual):
Sealed Bid
less Clock
(t-statistic)
Clock Auctions
Actual
91.9%
(0.83)
91.9%
(1.04)
90.6%
(1.29)
Predicted
92.7%
(0.96)
90.8%
(0.78)
99.8%
(0.08)
Difference
-0.8%
(1.18)
1.2%
(1.17)
-9.2%**
(1.29)
Actual
92.9%
(0.72)
89.0%
(1.05)
88.4%
(1.34)
Predicted
93.5%
(0.56)
98.9%
(0.19)
99.7%
(0.07)
Difference
-0.6%
(0.81)
-9.8%**
(1.08)
-11.4%**
(1.35)
-1.00
(-0.93)
2.90
(1.93)+
2.20
(1.17)
93.8%
(0.63)
94.0%
(0.78)
94.4%
(1.39)
92.6%
(0.70)
99.0%
(0.17)
100%
(0.00)
1.2%
(1.00)
-5.0%**
(0.84)
-5.6%**
(1.39)
94.0%
(0.62)
93.6%
(0.80)
95.4%
(1.20)
97.9%
(0.28)
99.2%
(0.15)
100%
(0.00)
-3.9%**
(0.70)
-5.7%**
(0.86)
-6.1%**
(1.20)
-0.20
(-0.15)
0.40
(0.39)
-1.00
(-0.52)
** Significantly different from 0 at the 1% level, 2 tailed t-test
TABLE 8
Revenue (in dollars)
(standard error in parentheses)
No.
Computers
vh
Value
$3.00
3
$4.00
$4.40
$4.00
5
$4.40
$5.10
Sealed Bid Auctions
Difference
(actual):
Sealed Bid
less Clock
(t-statistic)
Clock Auctions
Actual
8.48
(0.33)
9.33
(0.24)
9.60
(0.26)
Predicted
5.80
(0.12)
8.42
(0.09)
11.45
(0.24)
Difference
2.68**
(0.29)
0.91**
(0.23)
-1.85**
(0.30)
Actual
7.14
(0.24)
8.31
(0.23)
9.23
(0.31)
Predicted
5.84
(0.12)
9.93
(0.13)
10.81
(0.18)
Difference
1.30**
(0.24)
-1.62**
(0.24)
-1.58**
(0.26)
1.34
(3.38)**
1.02
(3.01)**
0.37
(0.88)
11.10
(0.17)
11.66
(0.17)
12.13
(0.16)
9.01
(0.12)
12.64
(0.14)
12.60
(0.15)
2.08**
(0.19)
-1.00**
(0.16)
-0.47**
(0.15)
9.85
(0.16)
10.88
(0.21)
11.78
(1.27)
10.16
(0.13)
11.72
(0.13)
12.37
(0.14)
-0.31
(0.16)
-0.84**
(0.19)
-0.59
(0.16)
1.25
(5.24)**
0.78
(2.78)**
0.35
(1.43)
** Significantly different from 0 at the 1% level, 2 tailed t-test.
Figure Caption
Figure 1: Bid patterns for the multi-unit demand bidder (h) in the sealed-bid auctions (top panel) and
the ascending price auctions (bottom panel). There are three bid regions:
Region 1: Allocations and prices are the same in both auctions as bidder practice demand
reduction.
Region 2: Prices and allocations differ as h never wins one unit in the ascending price auctions,
but does so at times in the sealed bid auctions. There is a potential exposure problem in region 2 in
both cases.
Region 3: Bidders “go for it,” winning both units for sure. The size of region 3 is smaller in the
ascending price auctions and, unlike the sealed-bid auctions, there are no potential exposure problems.
Appendix A: Sealed-Bid Uniform Price Auction with Synergies.
We derive theoretical predictions provided in Sections I and II of the text..
There are (n + 1) > 2 bidders and m = 2 units auctioned. The (n + 1)th
bidder, denoted by h, has a concave utility function u(π) that is normalized
so that u(0) = 0 and u0 (0) = 1 and where π represents earnings net of cost
of purchasing units. h demands two units valuing each, by itself, at u(V ).
Bidders 1, 2, ..., n, demand only one unit valuing it at V1 , V2 , ..., Vn , respectively.
V1 , V2 , ..., Vn and V are independent random variables from F (·) and Fh (·),
respectively on the common support [0, 1]. V(k) denotes the kth order statistic
of V1 , V2 , ..., Vn , and F(k) its distribution function. Let v1 , v2 , ..., vn and v, be
the realizations of V1 , V2 , ..., Vn and V and, without loss of generality, assume
that v1 ≥ v2 ≥, ..., ≥ vn . The good is only available in integer units. We are
interested in a sealed-bid uniform-price (highest losing bid) auction (SBUPA).
Bidders 1, ..., n, who demand a single unit have a dominant strategy to bid their
true value. Denote by p the price per unit h pays. Although the value of winning
a single unit for h is u(v − p), there is a supper addictive value for winning both
units. If h wins both units her utility is u(2v + g(v) − 2p), i.e., she is getting an
extra g(v), where g(0) = 0 and g0 (v) > 0. In the experiment g(v) = v, which we
assume throughout the appendix.
Without loss of generality assume that b1 (v) ≥ b2 (v) represents h’s two
(optimal) bids.
Lemma 1 (a) b1 (v) ≥ v.
(b) If b1 (v) > v, then b1 (v) = b2 (v).
Proof. (a) Suppose (a) does not hold. This implies that there exists v∗
such that v∗ > b1 (v∗ ) ≥ b2 (v∗ ). But then, raising b1 (v∗ ) from b1 (v∗ ) < v ∗ to
∗
∗
b#
1 (v ) = v makes h, better off when it matters since in such events h will
win one unit rather than zero, which results in strictly positive expected utility.
(b) Suppose (b) does not hold. This implies that there exists v∗ such that
b1 (v∗ ) > v∗ and b1 (v∗ ) > b2 (v∗ ). Case 1. b2 (v ∗ ) ≥ v∗ . In this case the pair
∗
∗
∗
∗
∗
∗
{b#
1 (v ) = b2 (v ), b2 (v )} dominates the alternative {b1 (v ) > b2 (v ), b2 (v )},
# ∗
∗
∗
∗
i.e., reducing b1 (v ) > b2 (v ) to b1 (v ) = b2 (v ) dominates. Here is the reason:
If h wins two or zero units, the proposed reduction in b1 (v∗ ) does not matter.
However, if h wins one unit then the price is at least v ∗ and with a positive
probability strictly higher than v ∗ , implying that E[u(v ∗ − p)] < 0. Therefor,
by reducing b1 to b2 h wins no units, which generates strictly positive expected
utility realtive to b1 > b2 and losing money on the single unit earned. Case
2. b2 (v∗ ) < v∗ . Using similar arguments we can show that the pair of bids
∗
∗
∗
∗
∗
∗
∗
∗
{b#
1 (v ) = v , b2 (v ) < v } dominate {b1 (v ) > v , b2 (v ) < v }.
Part 1: We start the analysis by assuming first that b1 (v) = b2 (v) and
thus by Lemma 1-(a), b1 (v) = b2 (v) ≥ v. In this case h’s maximization problem
becomes:
1
Rb
(A1) Max{ 0 nf (t)[F (t)]n−1 u(3v − 2t)dt + n(1 − F (b))[F (b)]n−1 u(v − b)}.
b≥v
The integral component of (A1) represents h’s expected utility from winning
two units, in which case n bids are below b. The second part of (A1) represents
h’s expected utility from winning one unit, an event where v1 , the highest rivals’
bid, is higher than b but all other bids are below b. In all other events h
earns u(0) = 0. The first order condition (FOC) for maximization of (A1) after
rearranging becomes:
(b)
1−F (b)
0
(A2) u(3v − 2b) − u(v − b) + (n − 1)u(v − b) 1−F
F (b) − u (v − b) f (b) = 0.
The left hand side (LHS) of (A2) evaluated at b = v is:
(A3) u(v) −
1−F (v)
f (v)
=: H(v).
We assume that H(v) is strictly monotonic in v.1
Lemma 2 There exists a unique v = vc , satisfying: (a) vc = b1 (vc ) = b2 (vc )
which solves the FOC (A2). (b) ∀v > vc , b1 (v) = b2 (v) > v. (c) ∀v < vc ,
b1 (v) = v > b2 (v).
Proof. (a) H(0) < 0, H(1) > 0 and by assumption H 0 (v) > 0.Thus, there
exists a unique vc , such that vc = b1 (vc ) = b2 (vc ) that solves the FOC (A2).
(b) ∀b = v > vc , the LHS of (A2) is strictly positive and the optimal bids are
b1 (v) = b2 (v) > v. (c) ∀b = v < vc , the LHS of (A2) is strictly negative. But
Lemma 1 implies that we cannot have b1 (v) = b2 (v) < v, and thus b1 (v) > b2 (v).
Further, from Lemma 1, b1 (v) ≥ v but a strict inequality implies (by the second
part of Lemma 1) that b1 (v) = b2 (v) > v a contradiction. We conclude that
∀v < vc , b1 (v) = v > b2 (v).
With a risk neutral (RN) h and after rearranging, equation (A2) becomes:
(b)
(A4) (v − b)[1 + (n − 1) 1−F
F (b) ] + v −
1−F (b)
f (b)
=0
In our work F (·) is a uniform distribution, H(vc ) = 0 implies vc = 1/2, and
equation (A4) becomes: (v − b)[1 + (n − 1) 1−b
b ] + v − 1 + b = 0, implying that,
(v − b)[b+ (n −1)(1− b)] +vb −b + b2 = (n − 1)b2 − {n + (n −3)v}b +(n − 1)v = 0.
we can write (A4) as a quadratic equation:
By denoting Φ(n, v) =: n+(n−3)v
n−1
1 McAfee and McMillan (1987, p. 708, fn.11) define J(v) =: v − 1−F (v) and showed that
f (v)
it must be increasing in v in order to have a well behaved bidding function in equilibrium.
J(v) is also referred to as virtual valuation by Myerson (1981), and as marginal valuation by
Bulow and Roberts (1989). The role that the monotonicity of J(v) plays for optimality and
efficiency of Independent-Private-Value auctions is well understood. Our H(v) coincides with
J(v) for a risk neutal h that we assume in the text but is “slightly” different otherwise.
2
(A5) b2 − Φ(n, v)b + v = 0.
Differentiating the LHS of (A5) with respect to b yields:
(A6)
∂
2
∂b {b
− Φ(n, v)b + v} = 2b − Φ(n, v).
The second order condition (SOC) for maximization requires that (A6), evaluated at the optimal b, is negative. Thus, 2b − Φ(n, v) < 0, implying that
2b2 −Φ(n,v)b+(v−v)
< 0, or finally, after using (A5):
b
(A7)
b2 −v
b
< 0.
We can write the solution to the quadratic FOC (A5) as:
(A8) b1,2 = {Φ(n, v) ± [(Φ(n, v))2 − 4v]1/2 }/2.
Note, that once [(Φ(n, v))2 − 4v] < 0, there is no solution to equation (A5).
It is easy to verify that since v ≤ 1, [(Φ(n, v))2 − 4v] is strictly decreasing in v
for all n ≥ 1. Let vcn , be the value of v that solves:
(A9) [(Φ(n, vcn ))2 − 4vcn ] = 0.
Thus, ∀v > vcn , [(Φ(n, v))2 − 4v] < 0, and the LHS of (A5) is strictly
positive implying that the optimal bid is b(v) = 1. Namely, for such (high) v’s,
h0 s optimal strategy is to “go for it,” bidding 1, winning two units for sure and
enjoying the synergy bonus, v. In what follows we restrict attention to v’s that
satisfy v ∈ (vc , vcn ] = ( 12 , vcn ].
The positive root of (A8) yields, b2 > (Φ(n, v))2 /2−v = [b+ vb ]2 /2−v, where
the last equality is obtain by using (A5). Thus, b2 −v > [b2 +2v+(v/b)2 −4v]/2 =
[b2 −2v+(v/b)2 ]/2 = [b−(v/b)]2 /2 > 0, which violates (A7). On the other hand,
by using (A6) and (A8) it is easy to verify that the negative root of (A8) yields,
2b − Φ(n, v) < 0, so that the SOC is satisfied. With some additional tedious
algebra one can verify that for ∀v ∈ ( 12 , vcn ], the negative root also yields b > v
as required. Thus, the negative root of (A8) satisfies the FOC, SOC and b > v,
and is rewritten as:
(A10) b = {Φ(n, v) − [(Φ(n, v))2 − 4v]1/2 }/2.
Although the solution proposed in (A10) satisfies the FOC and the SOC,
it assures only a local maximization since the objective function may not be
quasi-concave. Let E[π(b(v))] denote the expected payoffs for h who has V =
v ∈ ( 12 , vcn ] and is using b(v) as defined by (A10). E[π(b(v))] ={expected net
gain of winning two units}{probability of winning two units} + {expected net
gain of winning one unit}{probability of winning one unit}. Or formally:
3
2n
b(v)}{[b(v)]n }+{v−b(v)}{n[1−b(v)][b(v)]n−1 }
(A11) E[π(b(v))] = {3v− n+1
Let E[π(b ≥ 1)] denote the expected payoffs for h who has V = v ∈ ( 12 , vcn ]
and bids b ≥ 1 on both units which assures winning both of them:
(A12) E[π(b ≥ 1)] = [3v −
2n
n+1 ],
2n
as 3v is the value of winning two units and n+1
is the expected payment in
∗
this case. Let vn , be the v that equates equation (A11) and (A12). It turns out
that,
(A13) a) vn∗ ∈ ( 12 , vcn ], b) ∀v ∈ ( 12 , vcn ], the optimal bids are, b1 (v) =
b2 (v) = {Φ(n, v) − [(Φ(n, v))2 − 4v]1/2 }/2 and c) ∀v ∈ [vn∗ , 1] the optimal bids
are b1 (v) ≥ 1 and b2 (v) ≥ 1.
Part 2. Here we solve for the region where v < vc , implying by Lemma 1
and Lemma 2 part (c) that b1 (v) = v > b2 (v). Simplify by concentrating only
on b2 (v) and denoting it by b(v). Everything is the same as in part 1 with the
exception, that we first derive the results for any number of units auctioned,
m < (n + 1), and summarize them for our current experiment where m = 2 at
the end.
There are three regions (events) to consider here.2
Region 1: Here, V(m−1) ≤ b, thus, E[u(v, b)] =
Rb
0
u(3v − 2p)dF(m−1) (p).
Region 2: Here, V(m) ≤ b < V(m−1) , thus, E[u(v, b)] = u(v − b)[F(m) (b) −
F(m−1) (b)].
Region 3: Here, b < V(m) < v, thus, E[u(v, b)] =
Rv
b
u(v − b)dF(m) (p).
Region 1, is the event that h wins both units and earns u(2v +v −2p); region
2 is the event that h wins only one unit, and her bid, b, sets the price (which
affects her gains on the unit won); region 3 is the event that h wins only one
unit and does not set the price. We differentiate with respect to b and collect
terms from the three region to obtain the following FOC for maximization:
∂E[u(v, b)]/∂b = [u(3v − 2b))f(m−1) (b)] − {u0 (v − b)[F(m) (b) − F(m−1) (b)] +
u(v−b)[f(m) (b)−f(m−1) (b)]}−[u(v−b)f(m) (b)] = [u(2(vb))−u(v−b)]f(m−1) (b)−
u0 (v − b)[F(m) (b) − F(m−1) (b)],
where, f(k) ≥ 0 is the derivative of F(k) . Finally, using the fact that [F(m) (b)−
¡ n ¢
¡ n−1 ¢
F(m−1) (b)] = n−1
[1 − F (b)]m−1 [F (b)]n+1−m , and that f(m−1) (b) = m−2
[1 −
m−2
n+1−m
F (b)]
[F (b)]
f (b), we obtain:
2 The derivation here is similar to the derivation of the FOC for the multi-unit demand
SBUPA auctions with flat or decreasing demands developed in Ausubel and Cramton (1996),
and Kagel and Levin (2001).
4
(A14) [u(3v − 2b∗ ) − u(v − b∗ )] −
u0 (v−b∗ )(1−F (b∗ ))
(m−1)f (b∗ )
≤ 0,
where b∗ is the solution to the problem with a risk averse (RA) h, and where
strict inequality holds only if b∗ = 0.
Fact 1. When m = 2, as in our experimental design, when v < vc = 12 ,
b < v, so that even with synergies there is demand reduction on the second unit.
(v)
To see why, consider the LHS of (A14). At b = v it is equal to [u(v)− 1−F
f (v) )] <
∗
[v −
1−F (v)
f (v) )], since u(v) is concave, u(0) = 0
(v)
1−F (vc
vc = 12 , [v − 1−F
f (v) )] < [vc − f (vc ) )] =
and u0 (0) = 1. However, since
v <
0 (see the proof to Lemma 2).
Thus, the LHS of (A14) evaluated at b = v is strictly negative. Note from
(A14) that b∗ is independent of n, the number of single unit demanders, for all
(concave) u’s. This surprising result is reminiscent of the optimal reservation
price result in a single unit, independent-private-value auction. For the RN case
(A14) becomes:
(A15) (2v − b) −
1−F (b)
(m−1)f (b)
≤ 0,
with strict inequality only if the optimal b = 0. It is easy to verify that a
sufficient condition to assure quasi-concavity of the objective function for the
RN case is:
(A16) (1 − F (b))f 0 (b) − (m − 2)[f(b)]2 ≤ 0.
With a uniform distribution, f 0 (b) = 0 so that (A16) is satisfied for all
m ≥ 2.
We turn now to the effect of RA on bidding for the case of v < vc . Let
0
∗
)(1−F (b∗ ))
v > b∗ > 0. Thus, 0 = [u(3v − 2b∗ ) − u(v − b∗ )] − u (v−b
< [u0 (v −
(m−1)f (b∗ )
0
∗
∗
∗
)(1−F (b ))
1−F (b )
= u0 (v − b∗ )[(2v − b) − (m−1)f
b∗ )(2v − b)] − u (v−b
(m−1)f (b∗ )
(b∗ ) ], where the first
equality is just (A14) and the strict inequality is due to the concavity of u. We
1−F (b∗ )
conclude that [(2v − b) − (m−1)f(b
∗ ) ] > 0, which we use to establish the next
fact.
Fact 2: The effect of RA is to reduce the bid of h on the second unit,
bRA (v) < bRN (v), unless bRN (v) is already zero. That is, under condition (A16)
a RA h bids no more on the second unit than a RN h when bRN (v) = 0, and
strictly less when bRN (v) > 0. Quasi-concavity of the RN case and the fact
that the FOC for the RN h evaluated at b∗RA is strictly positive is sufficient to
establish fact 2.
In the RN case it is convenient to rewrite the optimal b (from A15) as:
(A17) b(v) =
½
2v −
0,
b∈
/ [0, v],
1−F (b)
(m−1)f (b) ,
¾
b ∈ [0, v]
5
,
which implicitly solves for the (optimal) b. In our design F (·) is uniform on
[0, 1] so that (A17) reduces when m > 2 to,
(A18) b(v) =
½
0,
¾
.
1
1
v ∈ [ 2(m−1)
,m
]
1
v ∈ [0, 2(m−1)
],
(m−1)2v−1
,
m−2
When m = 2, and v < vc = 1/2, the LHS of (A15) becomes [2v − 1] which
is strictly negative. Thus, establishing for a RN h, a uniform distribution and
m = 2 (as in our experimental design):
(A19) b(v) = 0, ∀v ∈ [0, 12 ).
Summary of Appendix A for our Experimental Design.
The n bidders with a single unit demand have a dominant strategy b(vi ) = vi
and are replaced by computers who employ this strategy and the human knows
this fact. In our experiment the number of units auctioned is always two, n is
either three or five, and g(v) = v (so that g(0) = 0 and g0 (v) > 0 and g 0 (0) = 1
are satisfied). The distribution function F (·) is uniform on [$0, $7.5].
Transforming the analysis above from the domain of F (·) on [0, 1] to [$0, $7.5]
is trivial. We end up with a partition of [$0, $7.5] to three regions.
Region 1 is [$0, $3.75). In this region the optimal bid is b1 (v) = v, b2 (v) = 0.
Region 2 is [$3.75, $7.5vn∗ ), where vn∗ equates equations (A11) and (A12) in
the appendix. In this region the optimal bid is b1 (v) = b2 (v) = $7.5{Φ(n, v) −
[(Φ(n, v))2 − 4v]1/2 }/2.
Region 3 is [$7.5vn∗ , $7.5]. In this region the optimal bids are b1 (v) ≥ $7.5,
b2 (v) ≥ $7.5.
6
Appendix B: English-Clock Auction (ECA) with Synergies.
In what follows we simplify notation by using v for vh . Also recall that the
synegy bonous is modeled as earning an extra g(v) = αv if h obtains both units.
The optimal strategy for h in the ECA can be nicely described by patitioning
the domain of values to three regions:
1
(B1) A =: [0, 1+α
),
1
2
B =: [ 1+α
, 2+α
),
2
C =: [ 2+α
, 1].
1
).
A. Optimal Behavior for h when V = v ∈ A =: [0, 1+α
A.1 If v3 ≥ v, drop the first unit at any price, p ∈ [0, v3 ] If v3 < v, drop
the first unit at any price, p ∈ [0, min{v, v2 }].
A.2 Drop the second unit at price, p ∈ [v, max{v, v3 }]
Proofs and observations:
Observation 1. In region A it is never optimal for h to win both units.
To win two units h must stay IN with both beyond clock price, p = v2 . By
dropping a unit at p = v2 , h stops the auction, “wins one unit” (WOU) and
earns, π(W OU ) = [v −v2 ]. Suppose that h decides to stay IN with both units an
extra δ beyond p = v2 , (as long as v2 +δ ≤ 1) and drop out at p = v2 +δ if V1 = v1
does not drop by then. Recall that given that V2 = v2 , V1 |V1 ≥ v2 is distributed
δ
, v1 drops within the next δ, h wins
uniformly on [v2 , 1]. With a probability 1−v
2
two units and earns: (2+α)v−2E[V1 |v2 +δ ≥ V1 ≥ v2 ] = [(2+α)v−2v2 −δ]. With
2 −δ
a probability of 1−v
1−v2 v1 does not drop in that interval and h stops the clock at
p = v2 +δ to win one unit and earn (v−(v2 +δ)). (Note that since the possibility
p = v2 +δ = 1 is allowed, we are also including the strategy that assures winning
two units.) Thus, expected profits for staying IN beyond v2 and “possibly
δ
2 −δ
[(2+α)v−2v2 −δ]+ 1−v
winning two units” are: π(W T U ) = 1−v
1−v2 [v−v2 −δ] =
2
δ
1
1−v2 [(1+α)v−v2 ]+[v−v2 −δ]. However, since v < 1+α , {π(W T U )−π(W OU )} =
δ
δ
1−v2 [(1 + α)v − v2 − (1 − v2 )] = 1−v2 [(1 + α)v − 1] < 0.
Observation 2. As a result of Observation1, h never wants to win even one
unit at a clock price, p > v, as it earns negative profits rather than zero with no
units won.
Rules A.1 and A.2 are the most general rules that implement these conclusions. Note that max{v, v3 } ≥ v ≥ min{v, v2 }. Also note that the requirement
p ∈ [0, min{v, v2 }], rather than p ∈ [0, v2 ], both of which assure winning no
more than one unit, is to avoid staying IN with two units beyond p = v, in
which case it is not desirable to win even one unit.
7
Note that in region A h’s optimal strategy yields the same prices and allocation as the strategy: ”Drop unit 1 at clock price, p = 0 and stay IN until the
clock price, p, reaches your value v.” Thus, in region A bidding yields the same
allocations and prices as in a sealed bid uniform price auction (see Appendix
A). Further, prices and allocations in region A are the same as when h has flat
or decreasing demand for the the second unit (Kagel and Levin, 2001).
Behavior outside equilibrium in region A: Dropping both units too
early leaves no further action. If h errs, staying active on the first unit longer
than is optimal (i.e., p > v) and then realizes his mistake, h ought to drop out
right away.
1
2
, 2+α
).
B. Optimal Behavior for h when V = v ∈ B =: [ 1+α
1
2
, 2+α
) in
Let the clock price p∗ =: [(2 + α)v − 1]. Note that since v ∈ [ 1+α
∗
∗
∗
region B, p − v = (1 + α) − 1 ≥ 0 > (2 + α)v − 2 = p − 1. Thus, v ≤ p < 1.
B.1 If v2 < p∗ , “Go All The Way,” (ATW).
B.2 If v2 ≥ p∗ , drop both units at clock price p ∈ [p∗ , max{p, v3 }].
Proofs and observations:
B.1 For any given realization V2 = v2 , this strategy yields profits, π(AT W ) =
2
= (2 + α)v − 1 − v2 ≥ (2 + α)v −
(2 + α)v − 2E[V1 |V2 = v2 ] = (2 + α)v − 2 1+v
2
∗
1 − p = 0. Winning one unit earns (at best) profits of, π(W OU ) = v − v2 .
1
[π(AT W ) − π(W OU)] = (1 + α)v − 1 ≥ 0, since v ≥ (1+α)
and is strictly pos1
itive with any v > (1+α) . Thus, ATW, dominates winning one unit or winning
none.
B.2 Following the strategy prescribed in B.2 yields zero units and zero
profits. Winning one unit earns v−v2 < 0 and staying IN beyond the clock price,
p = p∗ , to win two units earns (2+α)v −2E[V1 |v2 ≥ p∗ ] < (2+α)v −2E[V1 |v2 =
p∗ ] = (2 + α)v − 1 − p∗ = 0.
It is easy to show that staying longer, δ > 0, beyond p = max{p, v3 }, (as
long as p∗ + δ < 1) and dropping out if no one else drops out once the price
reaches max{p, v3 } + δ, yields negative expected profits.
Behavior outside equilibrium in region B. Case B.1 Dropping two units
too early leaves no further action. If h dropped the first unit too early h needs
to stay with the second unit no longer than clock price p = max{v, v3 }. Case
B.2 (This is the most interesting “out of equilibrium” behavior.) If h realizes
that she stayed IN with both units too late i.e., although p > max{p∗ , v3 } and
v2 > p∗ , then if v2 is still IN (v2 ≥ p), h must drop immediately and nothing
happens relative to the optimal policy. However, if v2 has dropped OUT already
(v2 < p), then h should “go all the way” in order to win both units.
8
2
, 1].
C. Optimal Behavior for h when V = v ∈ C =: [ 2+α
Following the strategy prescribed in C yields two units and earns positive
2
and E[V1 ] < 1.
expected profits of: (2 + α)v − 2E[V1 ] > 0, since v ≥ (2+α)
For any given realization V2 = v2 , this strategy yields profits of π(AT W ) =
(2 + α)v − 2E[V1 |V2 = v2 ] = (2 + α)v − (1 + v2 ). On the other hand, winning
one unit yields profits of π(W OU) = v − v2 . Thus, {π(AT W ) − π(W OU )} =
2
− 1 = 2+2α−2−α
=
{[(2 + α)v − (1 + v2 )] − [v − v2 ]} = (1 + α)v − 1 > (1 + α) 2+α
2+α
α
2+α > 0.
Dropping early and winning zero units earn zero profits. Thus, the prescribed
strategy is optimal.
Behavior outside equilibrium in region C: If h dropped both units
there is nothing to do. If h erred and dropped one unit she ought to stay IN
with the second unit as long as the clock price, p < max{v, v3 }, and drop the
second unit immediately when p ≥ max{v, v3 }.
9